Media captionPrince William and son Prince George waved to crowds before entering hospital

The Duchess of Cambridge has given birth to a girl, Kensington Palace has announced.

The princess - who is fourth in line to the throne - was "safely delivered" at 08:34 BST, the palace statement said.

The Duke of Cambridge, who was present for the birth of the 8lbs 3oz (3.7kg) baby girl, brought his 21-month-old son Prince George to visit his sister at St Mary's Hospital, west London.

The couple and their daughter will leave hospital on Saturday evening.

They will travel home to Kensington Palace, where Prince George has now returned.

In a statement, Kensington Palace said: "Their Royal Highnesses would like to thank all staff at the hospital for the care and treatment they have all received.

"They would also like to thank everyone for their warm wishes."

As he left the hospital briefly to pick up his son, Prince William told the waiting crowds the couple were "very happy".

Both Catherine and her daughter are "doing well", Kensington Palace said. The name of the baby will be announced in due course.

For live updates click here and for the royal baby special report, click here.

The Prince of Wales, who had earlier said he was hoping for a grand-daughter, and the Duchess of Cornwall said they were "absolutely delighted".

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An official notice was placed on an easel at Buckingham Palace - it will be there for 24 hours

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People are queuing outside Buckingham Palace for a glimpse of the easel

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Cameras are focused on the door of the Lindo wing

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Prince William plants a kiss on his son's head before taking him inside the hospital to visit his baby sister

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It is the first time Prince George has been officially seen in public in the UK since he left the same hospital as a newborn in 2013

The duchess had been admitted to the private Lindo Wing at St Mary's Hospital in Paddington, west London, at 06:00 BST.

The birth announcement came just after 11:00 BST.

The statement from Kensington Palace added: "The Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news."

The latest royal joins the line of succession behind her grandfather Prince Charles, father Prince William and brother Prince George, who was born at the same hospital in July 2013.

The princess is the Queen's fifth great-grandchild.

Outside the Lindo Wing: Henrietta McMicking, BBC News

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Crowds began gathering after it was announced Catherine was in labour

Hundreds of well-wishers have gathered outside St Mary's Hospital to catch a first glimpse of the new princess.

Some are seasoned royal watchers. Margaret Tyler, 71 from Wembley, north-west London, has been waiting for the last 11 days: "But no nights, I'm too old for that now".

Meanwhile, Ash decided to bring his young daughter Ankita to mark the historic event. "We moved to London from the US nine years ago," he said. "I wanted to go and see Will and Kate's wedding but my wife thought I was mad."

There has been lots of speculation about a name from small girls in the crowd, from Disney-inspired Belle and Ariel to Alex, Stella and Rose.

And everyone wants a selfie with Tony Appleton, who runs a care home in Chelmsford, Essex, but has been a town crier for 30 years and stood on the steps of the Lindo wing to announce the news.

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The Royal Navy posted this celebratory picture by the crew of HMS Lancaster

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The BT Tower in London welcomed the news

Prime Minister David Cameron tweeted: "Congratulations to the Duke and Duchess of Cambridge on the birth of their baby girl. I'm absolutely delighted for them."

Deputy Prime Minister Nick Clegg tweeted that he and his wife Miriam wished the family "all the best".

And Labour leader Ed Miliband wrote: "Congratulations to the duke and duchess on the birth of their daughter. Wishing them lots of joy and happiness - and hopefully some sleep!"

The Archbishop of Canterbury Justin Welby said: "It's wonderful news that the Duke and Duchess of Cambridge have a healthy new baby girl.

"Like many here and abroad, I'm delighted for the royal couple on this special day. May God bless them and both of their children with love, health and joy."

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Some fans of the Royal Family have been outside the hospital for days

A large number of journalists, photographers and broadcasters had gathered after the announcement that Catherine was in labour.

Following the announcement of the birth, standing on the steps of the Lindo Wing, unofficial town crier Tony Appleton told crowds: "We welcome with humble duty the new baby of their Royal Highnesses the Duke and Duchess of Cambridge.

Media captionUnofficial town crier Tony Appleton on the steps of the Lindo Wing after the birth was announced

"The princess is fourth in line to the throne."

BBC royal correspondent Daniela Relph said the decision to leave the hospital would ultimately be a medical decision, but the duke and duchess would be acutely aware of the disruption their presence brings to the hospital.

There has been much speculation about the princess's name, with Alice being the favourite with several bookmakers. Charlotte, Elizabeth and Diana have also been popular with punters.

Prince George's name was announced two days after his birth, which is a relatively quick turnaround compared with previous royal babies.

It was seven days before the name of a newborn Prince William was announced in 1982, and there was a wait of a month following Prince Charles's birth in 1948.

Media captionSimon Clemison reports on how the news was broken to the world

The new royal will not be overtaken in the line of succession by any future younger brothers.

Under new rules which came into force in March, male bias was removed from the succession rules.

She will also be known as Princess of Cambridge after a letters patent were issued by the Queen.

It declared: "All the children of the eldest son of the Prince of Wales should have and enjoy the style, title and attribute of royal highness with the titular dignity of prince or princess prefixed to their Christian names or with such other titles of honour".

And the princess will also be able to marry a Roman Catholic without losing her place in line to the throne.